(* RELOCATION MAP ***********************************************************)
coinductive isid: predicate rtmap ≝
-| isid_push: â\88\80f,g. isid f â\86\92 â\86\91f = g → isid g
+| isid_push: â\88\80f,g. isid f â\86\92 ⫯f = g → isid g
.
interpretation "test for identity (rtmap)"
(* Basic inversion lemmas ***************************************************)
-lemma isid_inv_gen: â\88\80g. ð\9d\90\88â¦\83gâ¦\84 â\86\92 â\88\83â\88\83f. ð\9d\90\88â¦\83fâ¦\84 & â\86\91f = g.
+lemma isid_inv_gen: â\88\80g. ð\9d\90\88â\9dªgâ\9d« â\86\92 â\88\83â\88\83f. ð\9d\90\88â\9dªfâ\9d« & ⫯f = g.
#g * -g
#f #g #Hf * /2 width=3 by ex2_intro/
qed-.
(* Advanced inversion lemmas ************************************************)
-lemma isid_inv_push: â\88\80g. ð\9d\90\88â¦\83gâ¦\84 â\86\92 â\88\80f. â\86\91f = g â\86\92 ð\9d\90\88â¦\83fâ¦\84.
+lemma isid_inv_push: â\88\80g. ð\9d\90\88â\9dªgâ\9d« â\86\92 â\88\80f. ⫯f = g â\86\92 ð\9d\90\88â\9dªfâ\9d«.
#g #H elim (isid_inv_gen … H) -H
#f #Hf * -g #g #H >(injective_push … H) -H //
qed-.
-lemma isid_inv_next: â\88\80g. ð\9d\90\88â¦\83gâ¦\84 â\86\92 â\88\80f. ⫯f = g → ⊥.
+lemma isid_inv_next: â\88\80g. ð\9d\90\88â\9dªgâ\9d« â\86\92 â\88\80f. â\86\91f = g → ⊥.
#g #H elim (isid_inv_gen … H) -H
#f #Hf * -g #g #H elim (discr_next_push … H)
qed-.
(* Main inversion lemmas ****************************************************)
-corec theorem isid_inv_eq_repl: â\88\80f1,f2. ð\9d\90\88â¦\83f1â¦\84 â\86\92 ð\9d\90\88â¦\83f2â¦\84 â\86\92 f1 â\89\97 f2.
+corec theorem isid_inv_eq_repl: â\88\80f1,f2. ð\9d\90\88â\9dªf1â\9d« â\86\92 ð\9d\90\88â\9dªf2â\9d« â\86\92 f1 â\89¡ f2.
#f1 #f2 #H1 #H2
cases (isid_inv_gen … H1) -H1
cases (isid_inv_gen … H2) -H2
(* Alternative definition ***************************************************)
-corec lemma eq_push_isid: â\88\80f. â\86\91f â\89\97 f â\86\92 ð\9d\90\88â¦\83fâ¦\84.
+corec lemma eq_push_isid: â\88\80f. ⫯f â\89¡ f â\86\92 ð\9d\90\88â\9dªfâ\9d«.
#f #H cases (eq_inv_px … H) -H /4 width=3 by isid_push, eq_trans/
qed.
-corec lemma eq_push_inv_isid: â\88\80f. ð\9d\90\88â¦\83fâ¦\84 â\86\92 â\86\91f â\89\97 f.
+corec lemma eq_push_inv_isid: â\88\80f. ð\9d\90\88â\9dªfâ\9d« â\86\92 ⫯f â\89¡ f.
#f * -f
#f #g #Hf #Hg @(eq_push … Hg) [2: @eq_push_inv_isid // | skip ]
@eq_f //
qed-.
+
+(* Properties with iterated push ********************************************)
+
+lemma isid_pushs: ∀n,f. 𝐈❪f❫ → 𝐈❪⫯*[n]f❫.
+#n elim n -n /3 width=3 by isid_push/
+qed.
+
+(* Inversion lemmas with iterated push **************************************)
+
+lemma isid_inv_pushs: ∀n,g. 𝐈❪⫯*[n]g❫ → 𝐈❪g❫.
+#n elim n -n /3 width=3 by isid_inv_push/
+qed.
+
+(* Properties with tail *****************************************************)
+
+lemma isid_tl: ∀f. 𝐈❪f❫ → 𝐈❪⫱f❫.
+#f cases (pn_split f) * #g * -f #H
+[ /2 width=3 by isid_inv_push/
+| elim (isid_inv_next … H) -H //
+]
+qed.
+
+(* Properties with iterated tail ********************************************)
+
+lemma isid_tls: ∀n,g. 𝐈❪g❫ → 𝐈❪⫱*[n]g❫.
+#n elim n -n /3 width=1 by isid_tl/
+qed.